To train for a race, Will begins by jogging 11 minutes one day per week. He increases his jogging time by 6 minutes each week. Write the general term of this arithmetic sequence, a... To train for a race, Will begins by jogging 11 minutes one day per week. He increases his jogging time by 6 minutes each week. Write the general term of this arithmetic sequence, and find how many whole weeks it takes for him to reach a jogging time of one hour. Read more

Steps to Solve

1. Identify the First Term and Common Difference

Will begins jogging at 11 minutes and increases his time by 6 minutes each week. The first term of the sequence, $a_1$, is 11 minutes, and the common difference $d$ is 6 minutes.

2. Write the General Term Formula

The formula for the general term of an arithmetic sequence is given by:

$$ a_k = a_1 + (k-1) \cdot d $$

Substituting the known values:

$$ a_k = 11 + (k - 1) \cdot 6 $$

3. Simplify the General Term

Now, simplify the equation:

$$ a_k = 11 + 6k - 6 $$ $$ a_k = 6k + 5 $$

4. Determine How Many Weeks to Reach 60 Minutes

To find out how many weeks it takes to reach a jogging time of 60 minutes:

Set the general term equal to 60:

$$ 6k + 5 = 60 $$

5. Solve for $k$

Subtract 5 from both sides:

$$ 6k = 55 $$

Now, divide both sides by 6:

$$ k = \frac{55}{6} \approx 9.17 $$

Since we need whole weeks, we round up to 10 weeks.